1 Foraging Model

The proportion of time an animal is in a feeding behavioral state.

Process Model

\[Y_{i,t+1} \sim Multivariate Normal(d_{i,t},σ)\]

\[d_{i,t}= Y_{i,t} + γ_{s_{i,g,t}}*T_{i,g,t}*( Y_{i,g,t}- Y_{i,g,t-1} )\]

\[ \begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix} \] \[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}} * \beta \text{ dive depth}_{i,g,t} \] The behavior at time t of individual i on track g is a discrete draw. \[S_{i,g,t} \sim Cat(\phi_{traveling},\phi_{foraging})\]

Dive information is a mixture model based on behavior (S)

\(\text{Average dive depth}(\psi)\) \[ \psi \sim Normal(dive_{\mu_S},dive_{\tau_S})\]

Dive profiles per indidivuals

Data Statistics before track cut

## # A tibble: 11 x 2
##    Animal     n
##     <int> <int>
##  1 131111    58
##  2 131115   170
##  3 131116   294
##  4 131127  1327
##  5 131128    12
##  6 131130    70
##  7 131132   198
##  8 131133  1237
##  9 131134   279
## 10 131136   964
## 11 154187   275
## [1] 4884

Data statistics after track cut

## # A tibble: 10 x 2
##    Animal     n
##     <int> <int>
##  1 131111    38
##  2 131115   170
##  3 131116   291
##  4 131127  1045
##  5 131130    59
##  6 131132   151
##  7 131133  1161
##  8 131134   246
##  9 131136   825
## 10 154187   269

sink(“Bayesian/Diving.jags”) cat(" model{

pi <- 3.141592653589

#for each if 6 argos class observation error

for(x in 1:6){

##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]

#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
phi[i,g,t,1] <- alpha[state[i,g,t-1]] 
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement in location
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)

#number of dives per step length    
#divecount[i,g,t] ~ dpois(lambda_count[state[i,g,t]])

}

#Final behavior state
phi[i,g,steps[i,g],1] <- alpha[state[i,g,steps[i,g]-1]] 
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])

#for each dive depth
#dive depth at time t
dive[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])T(0.01,)

#dive speed (max depth/duration)
duration[i,g,t,u] ~ dnorm(duration_mu[state[i,g,t]],duration_tau[state[i,g,t]])T(0.01,)

#Assess Model Fit

#Fit dive discrepancy statistics
eval[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])
E[i,g,t,u]<-pow((dive[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

dive_new[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])T(0.01,)
Enew[i,g,t,u]<-pow((dive_new[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

}
}
}
}

###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016

##Behavioral States

gamma[1] ~ dbeta(3,2)       ## gamma for state 1
dev ~ dbeta(1,1)            ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2] <- gamma[1] * dev

#Intercepts
alpha[1] ~ dbeta(1,1)
alpha[2] ~ dbeta(1,1)

#Probability of behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

#Dive Priors
#average max depth
depth_mu[1] ~ dnorm(0.01,0.0001)

#we know that foraging dives are probably 0.1km deeper,
#but are not more than 0.5 km deeper
forage ~ dunif(0.05,0.5)
depth_mu[2] <- depth_mu[1] + forage

#speed = depth/duration priors
duration_mu[1] ~ dnorm(5,0.01)
time_forage~dunif(0,60)
duration_mu[2] = duration_mu[1] + time_forage 

#Dive counts, there can't be more than about 20 dives in a 6 hour period.
lambda_count[1] ~ dunif(0,20)
lambda_count[2] ~ dunif(0,20)

#depth and duration variance
depth_tau[1] ~ dunif(0,500)
depth_tau[2] ~ dunif(0,500)
duration_tau[1] ~ dunif(0,500)
duration_tau[2] ~ dunif(0,500)

##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd

#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539

#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156

}"
,fill=TRUE)

sink()

##     user   system  elapsed 
##   96.638   24.985 1896.361

1.1 Chains

1.2 Temporal autocorrelation in foraging

1.2.1 Time foraging as a function of time

1.3 Simulate dive depth

1.4 Simulate dive durations

1.5 Distribution of average dive counts per step

2 Temporal Variation in Dive Behavior

## # A tibble: 6 x 6
## # Groups:   Animal, Track [1]
##   Animal Track  step           timestamp       dive  Behavior
##   <fctr> <chr> <chr>              <dttm>      <dbl>     <chr>
## 1      1     1     1 2016-04-29 03:35:38 0.11566667  Foraging
## 2      1     1     2 2016-04-29 12:25:24 0.08150000  Foraging
## 3      1     1     3 2016-04-29 15:45:06 0.06783333  Foraging
## 4      1     1     4 2016-04-29 23:37:30 0.05400000 Traveling
## 5      1     1     5 2016-04-30 03:50:02 0.06033333 Traveling
## 6      1     1     6 2016-04-30 11:12:00 0.02583333 Traveling

2.1 Diel

2.2 Month

3 Comparison with 2d movement model

##    user  system elapsed 
##   0.567   0.070 287.938

3.1 Chains

Where does the 3d predict foraging that the 2d misses?

Where does the 2d predict foraging that the 3d refines?

3.2 Posterior Checks

The goodness of fit is a measured as chi-squared. The expected value is compared to the observed value of the actual data. In addition, a replicate dataset is generated from the posterior predicted intensity. Better fitting models will have lower discrepancy values and be Better fitting models are smaller values and closer to the 1:1 line. A perfect model would be 0 discrepancy. This is unrealsitic given the stochasticity in the sampling processes. Rather, its better to focus on relative discrepancy. In addition, a model with 0 discrepancy would likely be seriously overfit and have little to no predictive power.

## # A tibble: 1 x 2
##   `mean(E)` `var(Enew)`
##       <dbl>       <dbl>
## 1 0.1362699    12.01542